The Bezrukavnikov–Kazhdan–Varshavsky surjectivity conjecture for the depth zero Bernstein center

Assume $G$ is split, semisimple, and simply connected over $F=\mathbb{F}_q((t))$. Let $LG$ be the loop group, $\mathfrak A^{\operatorname{Fr}}(LG)$ the Frobenius-equivariant Grothendieck-group construction defined in the paper, and let

$$\mathfrak A^{\operatorname{Fr}}(LG)\xrightarrow{\langle A^{\operatorname{Fr}}\rangle}\mathfrak Z^{\operatorname{Fr}}(LG)\longrightarrow Z^0(G(F))$$

be the composite induced by the geometric construction and the sheaf-function correspondence. The Bezrukavnikov–Kazhdan–Varshavsky surjectivity conjecture. This composite map is surjective.

The conjecture would show that the geometric construction supplies every element of the depth zero Bernstein center. It is proposed in the paper as a generalization of the cited conjecture of Bezrukavnikov, Kazhdan, and Varshavsky, and remains open here.